Is it possible to introduce the concept of bitopological spaces such as $(X,T_1,T_2)$ (introduced by J.C.Kelly see Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly) in the homotopy theory or homology theory?

In some sense, the (relatively new) field of directed algebraic topology represents an attempt to do this. This article by Marco Grandis includes a discussion (on page 8) of why bitopological spaces are too general to admit a good (directed) homotopy theory. 

